3.6 Pricing mechanism

For several grid optimization problems represented in the form of (1), e.g., economic dispatch or power scheduling, DSWM, PSwEV, etc., an electricity pricing mechanism is needed to drive the electricity trading in the grid. Interestingly, the optimal Lagrange multiplier associated with the equality constraint (2) is often regarded as the market-clearing price. That will be proved under mild assumptions in the following:

Theorem 2. For a sufficiently small but positive ρ, i.e., ρ ! 0, the optimal energy price in the considering smart grid converges to λ<sup>∗</sup>, the optimal Lagrange multiplier corresponding to the constraint (34) or equivalently (26), as k ! ∞.

Proof: For the OEM problems in smart grids, the convex functions fi ð Þ xið Þt in (1) and (25) represent the costs for power generations or for the consumers' satisfaction. Therefore, we can denote the following functions as the reward functions for agents:

$$\mathcal{W}\_i(\mathbf{x}\_i(t)) = p(t)\mu\_i \mathbf{x}\_i(t) - f\_i(\mathbf{x}\_i(t)) \tag{46}$$

where p tð Þ is the energy price at time slot t. In the ADMM reformulation (28) and (29), xið Þ<sup>t</sup> is replaced by <sup>P</sup><sup>k</sup>þ<sup>1</sup> <sup>i</sup> ð Þt . For simplicity, the time index is also dropped hereafter.

The optimal energy price in the system is achieved when the market is cleared, at which the marginal costs of all agents are the same. Hence,

$$\frac{\partial \mathcal{W}\_i(P\_i^{k+1})}{\partial P\_i^{k+1}} = 0 \; \forall i = 1, \ldots, M \tag{47}$$

with the total maximum output of 210 MW. The parameters of generators and demand units are taken from [26]. The average PV output curve, which is shown in Figure 2, is suitably scaled from the real PV data collected in New England [28]. Moreover, an average load profile taken from New England Independent System Operator [29] is utilized and properly scaled to obtain the time-varying upper bounds of demand units. This load profile has two demand peaks at 11:00 am and

A Distributed Optimization Method for Optimal Energy Management in Smart Grid

DOI: http://dx.doi.org/10.5772/intechopen.84136

8:00 pm, and the highest demand is at 8:00 pm, as seen in Figure 3.

Figure 1. IEEE 39-bus system.

Figure 2. PV output.

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In the proposed SDC-ADMM algorithm, this means

$$\frac{\partial f\_i(P\_i^{k+1})}{\partial P\_i^{k+1}} = p\mu\_i \,\forall i = 1, \dots, M \tag{48}$$

Substituting (48) into (35), we have

$$p\mu\_i + \rho \left(P\_i^{k+1} - X\_i^k + \mu\_i^k\right) = \overline{\lambda}^{k+1} \mu\_i \,\forall i = \mathbf{1}, \ldots, M \tag{49}$$

If ρ ! 0, we obtain from (49) that

$$p = \overline{\lambda}^{k+1} \tag{50}$$

Thus, as long as the proposed SDC-ADMM algorithm converges, the optimal energy price <sup>p</sup><sup>∗</sup> is equal to the optimal Lagrange multiplier <sup>λ</sup><sup>∗</sup> <sup>≜</sup> lim<sup>k</sup>!<sup>∞</sup> <sup>λ</sup> kþ1 . ■
